Defender Michael Morrison gave Reading a 22nd-minute lead when nodding in from a corner.
On the half-hour, midfielder Ovie Ejaria made it 2-0 with an angled drive from outside the area.
Against a poor Luton side, Reading added a third goal 11 minutes from time when substitute Garath McCleary coolly slotted home.
Bowen replaced Jose Gomes last month and had led his side to two victories and a draw in his opening matches in charge.
Luton, the League One champions last season, had lost their previous three league games and six of their past nine.
It was their first meeting with Reading since August 2008, when the hosts won 5-1 in a League Cup second-round tie at the Madejski Stadium.
Although Cranie struck his shot powerfully, Rafael made a fine save.
Reading replied with a mazy run from Richards, whose eventual 20-yard effort produced a superb save from keeper James Shea.
However, from the resultant corner, Reading went ahead as John Swift slung over the set-piece and centre back Morrison rose alone to beat Shea with a firm header.
Eight minutes later, Reading increased their advantage.
A fast break out of defence along the left flank saw Swift release Liverpool loanee Ejaria, who cut in and curled a precise shot past Shea.
Sam Baldock had a chance to stretch the lead further before half-time but Shea made a fine stop from his accurate attempt.
Luton offered little going forward and were pressed back again after the interval.
A poor defensive clearance found the unmarked Swift but he lost his composure and blazed high over the crossbar.
Reading continued to push forward, with Luton still unable to make much headway in attack.
The home side resumed control when Richards once again found space on the left wing.
His excellent cross was met by George Puscas at the far post but Shea made a superb save to keep out the Romanian's close-range header.
McCleary sealed it for Reading in the 79th minute when seizing on an error by Alan Sheehan before running through to calmly beat Shea.